From Relaxed Indexability to Exact Indexability: A $t$-Step Approach for Partially Observable Restless Bandits

2026-08-25

AI Summary

本文针对部分可观测的不安分多臂老虎机问题，提出了一种t步前瞻阈值策略来逼近Whittle指数，该方法通过有限时域价值迭代提高了决策边界的准确性。

Abstract

Whittle index policies offer a scalable method for restless multi-armed bandits, but under partial observability even determining the indifference subsidy at a single belief requires solving an infinite-horizon belief-state problem with no closed-form value function. Liu [10] addresses this difficulty by linearizing the unknown decision boundary, leading to a linear system and a closed-form approximate Whittle index. However, the resulting threshold uses only a one-step active--passive comparison and does not account for longer-horizon continuation values. We extend this framework to a \emph{$t$-step lookahead threshold policy}. For each subsidy $m$, the threshold is defined by the active-minus-passive advantage under $t$-step finite-horizon value iteration. At $t=1$, the threshold is $m$-independent and recovers the linear threshold of Liu [10]; for $t>1$, it becomes subsidy-dependent through the induced first-crossing structure and tracks the exact decision boundary more closely. The proposed algorithm does not require indexability as an input and includes an indexability verification. Under the original Whittle indexability, we prove that the $t$-step approximate Whittle index converges geometrically to the exact Whittle index, \[ |\widehat W_t(ω)-W(ω)|=O(β^t). \] Numerically, all 2,715 tested three-state instances are verified as indexable according to the proposed criterion. The P95 index error decreases from $2.18\times10^{-2}$ at $t=1$ to $8.93\times10^{-4}$ at $t=8$. In an exact-comparable instance with $β=0.9999$, $t=2$ already recovers the exact Whittle-index ordering. Moderate-depth threshold policies also outperform the one-step baseline and remain close to the optimal dynamic-programming benchmark, while runtime grows mildly with $t$.

Problem

• Restless Bandits
• Partial Observability
• Whittle Index
• Indexability

Innovation

• t-step lookahead threshold policy
• finite-horizon value iteration
• indexability verification
• geometric convergence